3.1834 \(\int \left (a+\frac{b}{x^2}\right )^3 \, dx\)

Optimal. Leaf size=34 \[ a^3 x-\frac{3 a^2 b}{x}-\frac{a b^2}{x^3}-\frac{b^3}{5 x^5} \]

[Out]

-b^3/(5*x^5) - (a*b^2)/x^3 - (3*a^2*b)/x + a^3*x

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Rubi [A]  time = 0.042375, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ a^3 x-\frac{3 a^2 b}{x}-\frac{a b^2}{x^3}-\frac{b^3}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^3,x]

[Out]

-b^3/(5*x^5) - (a*b^2)/x^3 - (3*a^2*b)/x + a^3*x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{3 a^{2} b}{x} - \frac{a b^{2}}{x^{3}} - \frac{b^{3}}{5 x^{5}} + \int a^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**3,x)

[Out]

-3*a**2*b/x - a*b**2/x**3 - b**3/(5*x**5) + Integral(a**3, x)

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Mathematica [A]  time = 0.00849875, size = 34, normalized size = 1. \[ a^3 x-\frac{3 a^2 b}{x}-\frac{a b^2}{x^3}-\frac{b^3}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^3,x]

[Out]

-b^3/(5*x^5) - (a*b^2)/x^3 - (3*a^2*b)/x + a^3*x

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Maple [A]  time = 0.008, size = 33, normalized size = 1. \[ -{\frac{{b}^{3}}{5\,{x}^{5}}}-{\frac{a{b}^{2}}{{x}^{3}}}-3\,{\frac{{a}^{2}b}{x}}+{a}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^3,x)

[Out]

-1/5*b^3/x^5-a*b^2/x^3-3*a^2*b/x+a^3*x

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Maxima [A]  time = 1.44265, size = 43, normalized size = 1.26 \[ a^{3} x - \frac{3 \, a^{2} b}{x} - \frac{a b^{2}}{x^{3}} - \frac{b^{3}}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^3,x, algorithm="maxima")

[Out]

a^3*x - 3*a^2*b/x - a*b^2/x^3 - 1/5*b^3/x^5

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Fricas [A]  time = 0.211238, size = 50, normalized size = 1.47 \[ \frac{5 \, a^{3} x^{6} - 15 \, a^{2} b x^{4} - 5 \, a b^{2} x^{2} - b^{3}}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^3,x, algorithm="fricas")

[Out]

1/5*(5*a^3*x^6 - 15*a^2*b*x^4 - 5*a*b^2*x^2 - b^3)/x^5

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Sympy [A]  time = 1.41336, size = 32, normalized size = 0.94 \[ a^{3} x - \frac{15 a^{2} b x^{4} + 5 a b^{2} x^{2} + b^{3}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**3,x)

[Out]

a**3*x - (15*a**2*b*x**4 + 5*a*b**2*x**2 + b**3)/(5*x**5)

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GIAC/XCAS [A]  time = 0.226127, size = 45, normalized size = 1.32 \[ a^{3} x - \frac{15 \, a^{2} b x^{4} + 5 \, a b^{2} x^{2} + b^{3}}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^3,x, algorithm="giac")

[Out]

a^3*x - 1/5*(15*a^2*b*x^4 + 5*a*b^2*x^2 + b^3)/x^5